Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ornld Structured version   Visualization version   GIF version

Theorem ornld 938
 Description: Selecting one statement from a disjunction if one of the disjuncted statements is false. (Contributed by AV, 6-Sep-2018.) (Proof shortened by AV, 13-Oct-2018.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
Assertion
Ref Expression
ornld (𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))

Proof of Theorem ornld
StepHypRef Expression
1 pm3.35 609 . . 3 ((𝜑 ∧ (𝜑 → (𝜃𝜏))) → (𝜃𝜏))
21ord 391 . 2 ((𝜑 ∧ (𝜑 → (𝜃𝜏))) → (¬ 𝜃𝜏))
32expimpd 627 1 (𝜑 → (((𝜑 → (𝜃𝜏)) ∧ ¬ 𝜃) → 𝜏))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385 This theorem is referenced by:  friendshipgt3  26648  ralralimp  40309  av-friendshipgt3  41552
 Copyright terms: Public domain W3C validator